When Rational Sections Become Cyclic: Gauge Enhancement in F-theory via Mordell--Weil Torsion

We explore novel gauge enhancements from abelian to non-simply-connected gauge groups in F-theory. To this end we consider complex structure deformations of elliptic fibrations with a Mordell--Weil group of rank one and identify the conditions under which the generating section becomes torsional. For the specific case of Z2 torsion we construct the generic solution to these conditions and show that the associated F-theory compactification exhibits the global gauge group [SU(2) x SU(4)]/Z2 x SU(2). The subsolution with gauge group SU(2)/Z2 x SU(2), for which we provide a global resolution, is related by a further complex structure deformation to a genus-one fibration with a bisection whose Jacobian has a Z2 torsional section. While an analysis of the spectrum on the Jacobian fibration reveals an SU(2)/Z2 x Z2 gauge theory, reproducing this result from the bisection geometry raises some conceptual puzzles about F-theory on genus-one fibrations.


Motivation and Summary
It has been well-established that F-theory [1][2][3] is an efficacious tool for the geometric engineering of non-perturbative string theory vacua in even dimensions. An F-theory vacuum is associated to an elliptically fibered Calabi-Yau variety, π : Y → B, where the complex structure of the torus above each point of B specifies the value of the axio-dilaton at that point in a Type IIB compactification on B. The requirement that the axio-dilaton value at each point can be glued together to form a global elliptic fibration is necessary for the consistency of such a vacuum. Any elliptic fibration [4] can be cast in the form of a Weierstrass equation (1.1) where [x : y : z] are the coordinates of an ambient P 231 . The condition that this elliptic fibration is Calabi-Yau is expressed through the bundles that the coefficients f and g are global sections of, with K B the canonical class of B. One significant sector of the geometric engineering of Ftheory vacua involves finding necessary and sufficient conditions on the coefficients f and g such that the vacuum has particular physical features. For instance, the vanishing orders of f and g (and ∆ = 4f 3 + 27g 2 ) along certain divisors in B indicate the presence of singular fibers, by the Kodaira-Néron classification [5][6][7], which are in direct correspondence with the non-abelian gauge algebra in the F-theory effective physics.
In this paper we are interested in studying under what circumstances the non-abelian gauge group, G, in the low-energy physics has a non-trivial fundamental group. As is known from [8][9][10], π 1 (G) is associated to the existence of torsion inside of the Mordell-Weil group of the elliptic fibration. The Mordell-Weil group is the group of rational sections of the elliptic fibration, which is a finitely generated abelian group [11], and the group structure comes from the fiberwise application of the elliptic curve group law.
The free part, Z r , of the Mordell-Weil group gives rise to a U (1) r symmetry in F-theory [2,3], and the torsion part, Γ, is related to the global structure of the non-abelian gauge group, 1 π 1 (G) ∼ = Γ . (1.4) In order to determine the requirements for the existence of torsion inside of the Mordell-Weil group, we shall explore deformations of geometries such that a free section becomes torsional. In other words, we are considering a family of elliptic fibrations with Mordell-Weil rank r + 1 without torsion, and whose central fiber is a fibration of MW-rank r and non-trivial torsion Γ = Z n . Without loss of generality one can restrict one's attention to r = 0, and we shall do so henceforth. Physically, the complex structure deformation from the generic to the central fiber of the family corresponds to the enhancement of a U (1) into a non-simply-connected gauge group with π 1 (G) ∼ = Γ. Indeed, the enhancement of U (1)s into non-abelian symmetries has been studied extensively in recent literature. There, the process involves tuning two rational sections to sit atop one another. This has proven to be very effective in constructing global F-theory compactifications with higher dimensional representations, most recently SU (3) models with a 6 representation [13] and SU (2) models with 4 representation [14].
In this article we provide another approach to enhancement by tuning a rational section not to collide with another, but to sit globally at a specific Γ torsional point of the elliptic fiber. A special case of such a tuning was discussed in [10]. This paper puts the idea on a The generic elliptic fibration with Mordell-Weil torsion can be generated by the following process.
1. One begins with an elliptic fibration with a rank one torsion-free Mordell-Weil group.
It has been shown in [15] that any elliptic fibration with a rank one Mordell-Weil group is birationally equivalent to a Weierstrass model (1.1) with specific forms of f and g. At this point one has determined the sufficient and necessary conditions on f and g such that the elliptic fibration has Mordell-Weil group Γ. It now remains to study the physics associated to this generic model, such as the non-abelian gauge group and how π 1 (G) acts if G is not simple. However, one subtlety quickly arises; while every elliptic fibration with a rank one Mordell-Weil group is birationally equivalent to the generic model written down in [15], the birational transformation does not necessarily preserve the canonical class. Thus the elliptic fibration that is constructed in step 1 above may not be Calabi-Yau, and hence may not be immediately amenable to F-theory. The effective physics in [15], and in most of the subsequent literature, for example [16][17][18][19][20][21][22][23][24][25][26], assumed that the generic model was itself Calabi-Yau, however recent work [13,14,[27][28][29][30][31] has begun to explore the physics of those models where the generic model is not Calabi-Yau. The difference between the two situations is controlled by the height, or "tallness" [31], of the rational section, which we review in section 1.2.
In this paper we compute the generic form of an elliptic fibration with Z 2 torsion that arises from a complex structure deformation of the general elliptic fibration with rank one It is known in examples that there is an intimate relationship between the torsion subgroup of an elliptic fibration and the multi-section geometry of a certain "dual" genus-one fibration [27], where this notion of duality has mainly been explored when the elliptic fibration can be written as a toric complete intersection [33,34]. Where it has been studied such a mapping exchanges the Tate-Shafarevich group, which is the group of genus-one fibrations with the same Jacobian fibration and no isolated multiple fibers, and the torsion subgroup of the Mordell-Weil group. In section 4 we apply the understanding of Z 2 Mordell-Weil torsion acquired in the rest of the paper to construct a non-toric genus-one fibration with a bisection whose Jacobian elliptic fibration has Mordell-Weil group Z 2 . This example raises important questions about F-theory on genus-one fibrations.

Review of Elliptic Fibrations with Rank One Mordell-Weil Group
In [15] a general form for an elliptic fibration with Mordell-Weil rank one was written down.
This model, occasionally referred to as the Morrison-Park model, is given by a Weierstrass form (1.1) with specialized coefficients, f and g: The where K B is the canonical class of B and D is a divisor class on B. Up to a choice of twisting bundle O(β), the classes associated to the specialized coefficients can be determined through the classes of f and g. First, it can easily be seen that the class of c 0 must be even, and so it can be fixed via and thus the rest of the classes follow, giving (1.10) Because the coefficients have to be globally well-defined sections of corresponding line bundles, the classes in (1.10) must not be anti-effective. This constraints the possible choices of classes β and D for a given base B.
Given the Weierstrass model (1.7) it is easy to see that there are generically two independent rational sections located at We note that the generic Weierstrass model in (1.7) is singular, in particular the fiber above the locus b = c 3 = 0 is a nodal rational curve. One can see the singular nature of the Weierstrass model by observing the discriminant of (1.7): (1.12) An alternate way to realize the elliptic fibration (1.7) is via a birationally equivalent, singular, hypersurface in a P 112 fibration over B: in the ambient P 112 fiber mark two distinct points on each elliptic fiber, and thus give rise to two distinct rational sections. Indeed such a hypersurface was determined in [15] to be the generic form for elliptic fibrations with two rational sections, using analogous arguments to those in [4] for deriving the generic Weierstrass equation for elliptic fibrations, which have one rational section. Such a construction can then be mapped into the specialized Weierstrass form (1.7) using Nagell's algorithm [36].
As discussed, for generic choices of coefficients b, c i , the section S Q generates a rank one Mordell-Weil (sub-)group. F-theory compactified on this space, or more precisely on a Calabi-Yau space in the same birational equivalence class, therefore has a U (1) gauge factor. The massless spectrum of this theory contains hypermultiplets which are charged under the U (1) gauge group, and the range of charges that arise generically depends on the divisor class D [31].
The requirement for the existence of certain charged states depending on D can be seen through the Néron-Tate height. The height of a rational section is the projection to B of the self-intersection of the divisor associated to the Shioda map, σ(S Q ), of the section [37,38]: This height, which is a divisor in B, is related to the anomaly of the U (1) gauge factor associated to the section, if B is a twofold base, and anomaly cancellation further relates the charges of the U (1) charged hypermultiplets to h(S Q ) [32]. Assuming that the model (1.7) has no codimension-one singularities except the generic type I 1 fibers, the height was worked out in [15], and further one can see that it is bounded by the requirement that β be an effective divisor class, From the height one can define a notion of so-called tallness [31], which is constrained by the inequality (1.16), where the product is understood as the intersection product on the twofold base B. Here the index I runs over the charged hypermultiplets, which have charge q I . It is found that the value of t(S Q ) fixes a minimal largest charge that is forced to appear in the model for a consistent, anomaly free theory.
We will be principally interested in computing the spectrum in the case (1.7) itself defines a Calabi-Yau elliptic fibration, in which case D = 0, and the bound (1.16) is saturated when for which the tallness of the section is As such, the theory is required to contain a hypermultiplet of charge at least 2, but not necessarily of any higher charge. Indeed if one studies the generic model (1.7) then one can observe that the matter spectrum of this theory, with D = 0, consists of charge 1 and 2 hypermultiplets [15]. In [27] it was argued that given any Weierstrass model with non-trivial Mordell-Weil generator S Q , the union of all singlet matter loci is given by the complete For the generic single U (1) model (1.7) that we consider, this is in agreement with the results in [15]: The ideal I has two associated primes, p 1 and p 2 = (b, c 3 ), with the charge 2 singlets localized at V (p 2 ), and the charge 1 singlets sit at V (p 1 ). It can be shown that in terms of cycle classes, Thus, the multiplicities of the charged singlets are given in terms of intersection numbers as: (1.23)

Gauge Enhancement via Z Torsion
In this section we will determine a Weierstrass fibration that is birationally equivalent to any elliptic fibration with Mordell-Weil torsion Γ = Z 2 . For the construction we assume, as discussed in section 1, that the elliptic fibration fits into a family of elliptic fibrations with a U (1) gauge group, however we do not require any constraints on the dimension of the elliptic fibration, or on its canonical class.
We begin by utilizing that a generic element of such a family of elliptic fibrations is birationally equivalent to a Weierstrass equation of the form (1.7). If the rational section, S Q , located at the point (1.11) in the Weierstrass model is to be situated globally at the Z 2 torsion point of the elliptic fiber, then one must, as has been determined in appendix A, satisfy as a globally valid equation.
From the ideal (1.21) giving the codimension two loci in the base at which the degenerate fibers, and thus the matter hypermultiplets, are located in the U (1) model we can see that one of the two generators is y Q . As such, it is evident that, after solving (2.1) the second equation of the ideal, defines a codimension one locus of degenerate fibers, which will in turn give rise to a nonabelian gauge algebra. We point out that the compensation for the loss of a U (1) gauge group by a non-abelian gauge group, G, is expected as the F-theory gauge algebra must have a center which contains a Z 2 such that it is consistent to have π 1 (G) = Z 2 . In the following, we will explicitly determine the non-abelian gauge group of the enhanced theory, which turns out to be more intricate than, perhaps, naively expected.

Deforming to Z 2 Torsion
To solve the tuning condition y Q = 0 as a globally valid equation, we examine y Q -which is a global section of some line bundle -locally, through the restriction of y Q to local rings of function germs O B,p . The assumption of a smooth base B implies that for any point p ∈ B, this local ring is a unique factorization domain (UFD) [39]. For the case at hand, where we wish to solve (2.1), we observe that where now r and s are coprime over the UFD. By direct substitution the polynomial (2.3) becomes The first solution of the tuning condition that would give rise to a torsional section would be if σ vanished globally, however in such an eventuality one can see that the discriminant, given in (1.12), also vanishes globally, and thus the putative elliptic fibration is everywhere degenerate. We must thus only consider solutions with the vanishing of the second factor in (2.5). The form of the second factor requires s 2 to divide r 3 , however since r and s are coprime this is only possible if s is globally constant. As such the first requirement that (2.3) holds as a global equation is that and the remnant equation that must be solved is The solution is generic in that this solution is the most general way to solve the equation 3) over a UFD. The classes of the s i are determined by (2.8) and the classes (1.10). They can be expressed in terms of K B , β, D, and a further, a priori arbitrary, class Σ: These classes must not be anti-effective in order for the s i to be globally well-defined. For a fixed choice of base B, β, and D this requirement constrains Σ in terms of K B , β, and D.
For example, if B = P 2 , and thus −K B = 3H, where H is the hyperplane class, and we further choose β = n H, D = 0, and Σ = k H then the effectiveness requirement is satisfied if (2.10) Note that the range of n is dictated by the effectiveness of the classes in (1.10).
There are a multitude of specialized solutions for the tuning of a Z 2 torsional section that arise when the generic solution (2.8) is applied with non-generic s i . One relevant specialized tuning, which will be explored in more detail in section 3, is After such a tuning one can see that the coefficients b and c 2 are now simply written in terms of their coprime decomposition, with common factor s 3 . One can in addition seek the constraint that b and c 2 be generic divisors in B, which requires that the intersection of the two divisors be in codimension ≥ 2; this implies that there is no common component, or that s 3 does not vanish anywhere along B. For s 3 to be a constant function on B it is necessary that it transform as a section of O B . This is fixed, in addition to s 1 and s 4 being of the same classes as, respectively, b and c 2 , by imposing This particular specialization, whose explicit resolution will be studied in section 3, can be said to correspond to the generic solution (2.8) with the additional conditions that As such this specialized solution merely corresponds to setting with b, c 2 , and c 0 generic. There are many further specializations which change the structure and configuration of the singular fibers in the elliptic fibration whilst retaining the required Z 2 torsional section, however as they are all specialized solutions of (2.8) they shall not be explicitly considered further here.

F-theory of the Z 2 Torsional Model
In this section we elaborate on some of the effective physics of an F-theory compactification on the generic elliptic fibration with Z 2 Mordell-Weil torsion, as given through the Weierstrass elliptic fibration (1.7) with (2.8). In this we are hampered if the elliptic fibration has nontrivial canonical class, and thus for simplicity, we consider F-theory compactifications to 6D on elliptic Calabi-Yau threefolds of the specified form. The restriction to Calabi-Yau is equivalent to taking the divisor class D to be trivial in (1.10) and (2.9). The advantage of 6D compactifications is that there are strong anomaly conditions [32] with which we can bootstrap the spectrum without an explicit resolution.
If we plug the generic solution (2.8) into the expressions for f and g in (1.7), we obtain:   Finally, we also have the residual discriminant ∆ res = s 2 1 s 4 2 +4 c 0 s 2 3 −4 s 2 2 s 3 s 4 , which supports I 1 fibers, but no gauge symmetry. We note that the I 4 fiber may, at the level of the vanishing orders, not give rise to an su(4) gauge algebra but instead contribute an sp(2) gauge group from monodromy effects along the divisor s 3 [41][42][43].
Potential matter sits at codimension two loci where irreducible components of the discriminant intersect each other or self-intersect, and consequently the singularity type of the fiber enhances. Explicit computations reveal the irreducible codimension two loci, with corresponding vanishing orders of f, g, and ∆, summarized in table 1. The table also contains the matter representations, the origin of which we will discuss now in more detail.
Codimension Two Locus ord(f, g, ∆) Fiber Type Matter Representation  we study the branching rule for the adjoint representation of the gauge algebra associated with the enhanced singularity type into the product algebra of the colliding codimension one divisors. Hence, at the intersection locus {s 3 }∩{s 1 } of the su(4) and su(2) B divisors we locally have an so (10), and thus we expect matter in the (1, 6, 2) representation.
At the I * 0 enhancement over {s 4 } ∩ {s 1 }, the local algebra, by observing the vanishing orders, is so (8). Since this is an ordinary double point of the su(2) A divisor {t}, which is also (8). It is well-known [44,45] that an su(n) self-intersecting in an ordinary double point gives rise to the symmetric and antisymmetric representations 4 of the su(n).
Since there is an additional transverse su(2) algebra intersecting the self-intersection point the total matter representation here 5 expected is where the symmetric and anti-symmetric representations of su(2) A are just the adjoint and trivial representations, respectively.
In order to determine the multiplicities of the matter representations, we will impose the cancellation of all 6D gauge anomalies. It turns out that this uniquely fixes all multiplicities to be those shown in table 2. For more details of the anomaly cancellations in 6D F-theory compactifications see e.g. [46,47].
If we let x R ij denote the number of matter hypermultiplets in a representation R localized along the codimension two points at the intersection of the divisors D i and D j then we can make the ansatz In such an ansatz we are careful to distinguish the same representations, for example the two different (1, 1, 2), that arise from distinct pairs of intersecting divisors, and may have different coefficients n R ij from each codimension two locus. Furthermore, we have non-localized adjoint matter arising as deformation moduli of the gauge algebra divisors. These are counted by the geometric genus p g of the divisor [48]. For a smooth divisor D, the geometric genus agrees with the arithmetic genus 6 If D has singularities at points P k ∈ D, then the two genera differ by the delta-invariants associated with the singularities: For an ordinary double point singularity, which is precisely the singularity of the su(2) A divisor we are considering, the delta-invariant is δ k = 1. The exact multiplicity of the non-localized adjoint matter being the geometric genus follows from anomaly cancellation, and thus there are no n R ij parameters for this matter.
Inserting the multiplicities into the anomaly cancellation conditions for all three gauge factors, it turns out that the anomalies are canceled (independently of the choices for β, Σ) if 5 A similar situation arose in [14]. There, it was a single su(2) divisor which had a triple self-intersection. The conclusion is that one expects the trifundamental under the local su(2) ⊕3 algebra. In [14], since all three local su(2) copies were identified globally, the trifundamental decomposes into 2 ⊗ 2 ⊗ 2 = 2 ⊕ 2 ⊕ 4. 6 This formula holds of course only for divisors, i.e., curves, on a twofold base B.

Fiber Type
Matter Multiplicity where we have used the subscript h to distinguish the two different origins of (1, 1, 2) matter, consistent with table 1. The fact that the coefficients n R ij are 1/2 in some instances indicates that these are half-hypermultiplets that are situated at those particular codimension two points. As can be readily observed the representations associated to the half-hypermultiplets are all pseudo-real, and thus the half-hypermultiplet exists as a consistent state. The multiplicities of the matter are summarized in table 2.
Up until now, we have not discussed how the presence of the Z 2 torsional section affects the F-theory physics. A perhaps naive expectation, based on models in previous works [8] and [10], is that all non-abelian gauge factors should be affected by the Z 2 section, i.e., the global structure should be [SU (2)×SU (4)×SU (2)]/Z 2 . This would be consistent with the fact that -other than adjoints -only either bifundamentals of su(2) A ⊕ su(4), or matter carrying the anti-symmetric (6) representation of su(4) exist. However, the quotient structure should forbid (1, 1, 2) matter states, i.e., pure fundamentals under su(2) B . Given that we have these states, we propose that the global gauge group is In addition to the spectrum in table 2, this observation is also supported by the fact that the torsional section passes through the fiber singularities over the su(2) A and su (4)  To explicitly verify the global gauge group structure through homology relation as in [10] would require a global resolution of the model. While we will not attempt a resolution of the full model, we will present a resolution for a specialized case in section 3 that exhibits a More precisely, we will consider the resolution of a specialization of (2.15), corresponding to The spectrum of that model will then be determined explicitly, and summarized in table 3. It can be easily seen to match the result in table 2 upon imposition of the condition  Using an adjoint field, one can break G to its Cartan subgroup, U (1) 5 . There is then a large set of possible ways to break four out of the five U (1)s and obtain a spectrum with only the desired charged hypermultiplets. One must give vacuum expectation values to four of the many remaining fields, in such a way as to leave behind only a single U (1) gauge factor and no remnant discrete symmetries after the Higgsing. An effective approach is to make use of the Smith normal form [49] to keep track of these subtleties, as well as being implementable algorithmically. We exhaustively scanned through all of the possibilities, similar in spirit to the analysis performed in [50], and we find that all the Higgsing chains leading to such a spectrum fall into three distinct classes, associated to distinct twisting line bundles, O(β),

Resolution of Restricted Model and the Torsional Section
In the previous section the structure of the non-simply-connected gauge group in the effective physics was inferred by observing that the consistency of the matter spectrum would require that only the su(2) A and the su(4) could be quotiented by Z 2 . One can determine explicitly the action of the Z 2 by studying the crepant resolution of singularities of the Weierstrass model (2.15). Crepant resolutions in F-theory [51][52][53][54][55][56] allow one to observe physical features which  (1.13). On the right: The dual polygon, giving rise to the resolved hypersurface equation (3.1). As pointed out in [15], by assuming a unit coefficient in front of the term s w 2 , the two red monomials can be absorbed by a shift of w by a multiple of u.
are hidden in the singularities of the Weierstrass model; in this case we will see explicitly the torsional relation in homology that is induced by the torsional section.
In the following, we will present the resolution of the restricted version (2.27)  divisor and the fact that the other part of the non-abelian gauge group, su(2) B , is not affected by the torsional section. As we will see, these features can be directly extracted from the resolved fiber structure.

Toric Resolution
It was shown in the appendix of [15] that the codimension two singularities in the U (1) model (1.13) can be resolved torically. The introduced blow-up divisor, denoted s, vanishes precisely at the rational section in that model. We can write such a model as a hypersurface Y in a Bl 1 P 112 fibration X over the base B, given by the equation Such a singularity of Y can be resolved by further blowing up the ambient space at w = s = 0, introducing a coordinate γ, corresponding to a small resolution of the Calabi-Yau hypersurface. After the blow-up the elliptic fibrationŶ is given by the hypersurface equation in the blown-up ambient spaceX with SR-ideal The discriminant of this fibration will be useful later and is given by where the component b gives rise to the su(2) B algebra and (c 2 2 − c 0 b 2 ) to the su(2) A . The blow-up γ can be also engineered torically. In terms of the toric diagram of the fiber ambient space, this blow-up precisely corresponds to introducing an additional ray between the rays of w and s, as one can see in figure 2. This removes a vertex of the dual polygon, effectively setting c 3 = 0 in (3.1) and defining a new hypersurfaceŶ F 8 . The blow-up γ defines a section Λ = [γ] which generates a U (1) in the F-theory compactification onŶ F 8 . Thus, we can understandŶ as a non-toric restriction of the generic toric hypersurface 7Ŷ F 8 by c 1 → 0. It can be easily shown that this tunes the section Λ to be Z 2 torsional, thus enhancing the U (1) to a non-abelian symmetry.

Torsional Section and Global Gauge Group Structure
With the fully resolved elliptic fibration (3.5), we want to explicitly determine the homological relation that leads to the non-trivial global gauge group structure. First, it is straightforward 7 That is, generic up to the constant coefficient of the γ 2 s w 2 term.  As discussed in [27], this elliptic fibration has an I 2 locus above b = 0. Were it not for a unit coefficient in front of γ 2 s w 2 (which allows us to absorb the red terms), there would be another I 2 locus present. The non-toric tuning c 1 → 0 enhances the U (1) "torsionally".
to verify the su(2) B symmetry localized over b = 0. Over this locus, the resolved hypersurface (3.5) factorizes asP which is by definition the remainder of the total divisor class of (3.8) after the exceptional divisor corresponding to the affine node has been subtracted off.
The fiber splitting over the su(2) A locus can be described through prime ideals. Specifically, one of the two prime factors of the ideal (P , b 2 c 0 − c 2 2 ) is generated by four polynomials, This codimension two subvariety V (I) of the ambient spaceX is a divisor ofŶ localized over The intersection of this subvariety with u = 0 would require w = 0 as well, however, since u w is in the SR-ideal (3.6) this intersection is empty. In other words, the zero-section, U , does not intersect V (I), which hence corresponds to the non-affine Cartan divisor of su(2) A . Its homology class can be extracted using prime ideal techniques (see, for example, the appendix of [57]), yielding where ·X denotes the intersection product inX, and K B now abusively denotes the pullback of the canonical class of B toX. In terms of the ambient space homology, one can now use the linear equivalence and SR-ideal relations to show that By another abuse of notation, we will use the same label for (toric) divisors of the ambient spaceX and their pull-backs to the hypersurface. Then, the above equation implies that, in the homology ofŶ , we have This relation is the origin of the non-trivial global structure of the su(2) A factor [10], which we will review briefly for the case at hand. Suppose we have matter states w from M2-branes wrapping a fibral curve C in the elliptic fibrationŶ . Because of the relation (3.13), the Cartan where · now denotes the intersection product onŶ . Since C is a fibral curve (i.e., localized over a point in the base), the intersection with the pullback of a base divisor, like K B , vanishes.
We are left with the conclusion that C · (Λ − U ) = − q 2 . Now C, Λ and U -being classes of subvarieties ofŶ -are integral in homology. And sinceŶ is smooth by construction, its intersection pairing must be integral, forcing q/2 to be an integer. This implies that we cannot have any representations with odd charges under the su(2) A Cartan generator. In other words, the global structure of this gauge factor is SU (2)/Z 2 ∼ = SO(3). The torsional homology relation (3.13) does not involve the su(2) B divisor, so we have no such a restriction on the allowed representations of su(2) B . Hence, we find that the geometry of the resolved elliptic fibration (3.5) explicitly accounts for the global gauge group structure SU (2)/Z 2 × SU (2).

Matter States and Codimension Two Enhancements
We proceed with analyzing the matter enhancements and confirm the previous results (cf. table 2 with the condition (2.28)) found through anomaly cancellation. From the discriminant (3.7), we immediately see that the singularity enhances over the codimension two loci: Over the first locus resides a type III fiber, consisting of two fiber P 1 components intersecting each other in a double point. In F-theory, a codimension two enhancement from I 2 to type III hosts no matter.
The second locus, b = c 0 = 0, lies on the su(2) B divisor b = 0. Here, we find an enhancement to I 3 , and thus we expect fundamentals of su(2) B that are uncharged under su(2) A .
Concretely, setting b and c 0 to zero inP yieldŝ Note that the factorization of the quadratic term into P 1 ± involves taking the square root of c 2 in codimension two, which is generic on a twofold base. It is then straightforward to compute the intersection numbers with the divisor E B associated with the Cartan generator of su(2) B .
One readily finds Considering the SR-ideal (3.6), it is straightforward to show that the components intersect each other in an affine so(8) Dynkin diagram with one external node removed, as can be seen in figure 3. The component P 1 γ is the central node with multiplicity two, and P 1 s is the affine node intersected by the zero-section. These kinds of non-Kodaira fibers have been observed before and can be understood by studying the Coulomb branch of the associated M-theory compactification on the resolved geometry [58][59][60][61]. In [59], it was noted that the non-Kodaira singular fibers in codimension two have the form of contractions of Kodaira fibers (see also [62]), and this is consistent with what is observed here. A key point is that this particular fiber, where one specific node is deleted is related to the choice of resolution; Kodaira fiber where one multiplicity one node is removed.
topologically distinct crepant resolutions give rise to contracted I * 0 fibers with different nodes removed.
In order to compute the Cartan charge of the fiber components under su(2) A , we will use the result (3.13) of the previous subsection, With that, we can easily compute intersection numbers in the ambient space homology to

(3.20)
Therefore, omitting the effective curves whose homology class is not localized in codimension two, the full set of effective genus zero fiber curves that are localized at b = c 2 = 0 can be summarized as follows: Curve Cartan Charges Representation Note that both combinations P 1 γ +P 1 ± have the same Cartan charges, and it is a matter of choice into which representation we put each; however, full gauge invariance requires that one is in (3, 2) and one in (1, 2). 8 It can be observed that for each representation, the number of corresponding effective curves is half the dimension of the representation. This reflects the Locus Fiber Type Matter Rep. Multiplicity of Hypermultiplets Table 3: Representations and multiplicities of the matter associated to codimension two singularities for the restricted model (2.27).

Cancellation of Gravitational Anomaly
In section 2.2, we have used gauge anomalies to determine the spectrum. Here, we will discuss the cancellation of the gravitational anomaly; in anticipation of the discussion in section 4 we shall include several explicit details. In contrast to gauge anomalies, the gravitational anomaly is sensitive to uncharged hypermultiplets. For F-theory compactifications on a smooth elliptic Calabi-Yau threefoldŶ → B, the number of uncharged hypermultiplets is 1 + h 2,1 (Ŷ ). To compute h 2,1 (Ŷ ), we employ the standard relation The topological Euler characteristic χ top can be determined by dividingŶ into subspaces and using the additive property of χ top . This simplifies drastically if we can choose the subspaces such that they are all product spaces, in which case the Euler characteristic just becomes the product of χ top for the factors. For the elliptic fibrationŶ , the singular fibers in table 3 provide a natural division ofŶ into subspaces [47,63,64]. The dramatic simplification that occurs when considering an elliptic fibration can be summarized by noting that that is, the generic fiber, which has a smooth torus, or I 0 fiber, has vanishing Euler characteristic. Due to this the only subspaces that contribute to the Euler characteristic are those which involve the singular fibers. Specifically, the decomposition of the Euler characteristic is in terms of the following two classes of contributions.

Codimension two
Here the subspaces are of the form pt × fiber, so that For singular fibers consisting of smooth P 1 s with normal crossing intersections, the Euler characteristic can be computed by adding the contributions of each P 1 , which is 2 minus the number of intersection points on that P 1 , and then add the total number of intersection points in that fiber. For our model, we have type III fibers with χ top = 3, I 3 fibers which have χ top = 3, and the reduced I * 0 fibers with χ top = 5. Thus, the total contribution of codimension two fibers to χ top (Ŷ ) is the number of points in the base with these specific fiber types, see

Codimension one
The codimension one subspaces are ruled surfaces of the form Σ i × fiber i , where Σ i are the discriminant components with fiber type i. Thus the contribution to the Euler characteristic from these singular fibers is The topological Euler characteristic of Σ is given by The points P s are the codimension two enhancement points, of fiber type s, that have already been accounted for, and give rise to the correction term s s · #(P s ). The value of s depends on the singularity structure of Σ i at P s in the base [63]. For the case at hand, in table 3, we note that = −1 for the enhancement points of type III and I 3 fibers on any affected discriminant component. On the other hand, the coefficient for the I * 0 enhancement point depends on the divisor Σ for which we are considering the contribution: For Σ B = {b}, we have I * 0 = −1, whereas for Σ A = {c 2 2 − c 0 b 2 } the ordinary double point on the divisor gives I * 0 = 0. As a last ingredient, we need that χ top = 1 for singular I 1 fibers. In summary, we have the following codimension one contributions: (3.28) Thus, the Euler characteristic ofŶ , which is the sum of (3.25) and (3.28), is To employ the relationship (3.22) between χ top (Ŷ ) and the Hodge numbers, we now only need to know h 1,1 (Ŷ ), which by the Shioda-Tate-Wazir theorem [65] is where we have used that for a twofold base B, h 1,1 (B) = 10−K 2 B . This determines the number of uncharged hypermultiplets: Meanwhile, the number of charged hypermultiplets is given in table 3, and a quick counting The final contributions to the anomaly come from the six vector multiplets of the su(2)⊕su (2) gauge fields, and n T = h 1,1 (B) − 1 = 9 − K 2 B tensor multiplets 10 from divisors in the base. Thus, we verify that the gravitational anomaly cancels [46]:

Mordell-Weil Torsion in the Presence of Bisections
It has been observed in examples [27,33,34] that an elliptic fibration, Y , with Mordell-Weil torsion Z n is "dual" to a genus-one fibration Y ∨ with an n-section. To such a multi-section geometry, one can associate the Tate-Shafarevich group, X(J(Y ∨ )), consisting of the set of all genus-one fibrations, without isolated multiple fibers, which share the same Jacobian fibration [70][71][72]. For a genus-one fibration Y ∨ with an independent n-section and no codimension one singularities, is believed to encode the discrete Z n symmetry of F-theory compactified on J(Y ∨ ) [21,28,73,74]. In the presence of codimension one singularities, one observes that depending on the non-abelian gauge algebra, the discrete symmetry can be enhanced by the center [27,33,75]. In any case, the common folklore is that while the M-theory is different, F-theory compactified on any genus-one fibration in X(J(Y ∨ )) gives the same field theory as J(Y ∨ ). Thus, the conjecture is that for an F-theory compactification on Y with non-trivial global gauge group G/Z n , there is a dual compactification on an n-section geometry Y ∨ .
So far, the conjecture [33] is based on a set of toric examples [27,76]. For some of these there is a dual heterotic description [34], where this duality can be understood rigorously.
In these examples, the duality manifests itself as a "fiberwise mirror symmetry": Y and Y ∨ are generic complete intersections in an ambient space X resp. X ∨ , which are fibrations of a toric fiber ambient space A resp. A ∨ that are mirror to each other (i.e., they have dual toric fans).
However, it is currently not known how to generalize the duality to non-toric examples, mainly because there are no known such constructions.
In this section, we provide evidence that a model relevant in this context arises by deforming the geometry discussed in section 3. In particular, we propose that our non-toric construction yields a bisection geometry Y b and an associated Jacobian fibration J(Y b ), whose F-theory compactification has gauge group SU (2) Having both a bisection and Z 2 Mordell-Weil torsion in the Jacobian, the construction may be "self-dual" in the above sense, although we will not explore this direction in the present work. 11 Nevertheless, a better understanding of this model might shed light on a non-toric formulation of the duality. However, it turns out that just interpreting F-theory on the pair ( is more intricate than expected. In the following, we will see that such an interpretation will require further conceptual understanding of F-theory compactifications on multi-section geometries and their associated Jacobians.

A Jacobian Fibration with Torsional Section
It is well known [73-75, 77, 78] that the Morrison-Park model can be deformed through a conifold transition, which physically breaks the U (1) to a Z 2 symmetry by giving states with charge 2 a non-zero vacuum expectation value. In the Weierstrass form (1.7), the deformation b 2 → 4 c 4 yields a new Weierstrass model 11 Note that such self-dual examples also exist in the list of toric models [33].
This elliptic fibration has non-trivial Z 2 torsional three-cycles [74], which indicates the existence of a discrete Z 2 symmetry already in the M-theory compactification, and which uplifts to the F-theory compactification. Likewise, there are terminal singularities in codimension two of the fibration, which corresponds to matter charged only under the Z 2 [63,73,79,80].
This geometry is the Jacobian J(Y Z 2 ) of a generic hypersurface Y Z 2 in a P 112 fibration: While it is not instructive to present explicitly the discriminant and the enhancement loci, we do highlight that F-theory on J(Y Z 2 ) (which is the same as F-theory on Y Z 2 ) contains One can now deform these two geometries in an analogous way to that in section 3, namely by setting which in section 3 engineered a Z 2 Mordell-Weil group. In this case such a tuning results, as we will momentarily see, in Z 2 Mordell-Weil torsion when applied to the Jacobian fibration, J(Y Z 2 ), but not in the genus-one bisection fibration, Y Z 2 , as the notion of the Mordell-Weil group exists only for elliptic fibrations 12 . Note that this tuning can not be torically realized in the P 112 hypersurface (4.4), as the tuning (4.6) does not correspond to removing vertices of the dual polygon (see, e.g., figure 5 in [15] for a description in the same notation).
The resulting hypersurface Y b in the P 112 fibration, remains a smooth genus-one fibration with a bisection, thus we expect For the Jacobian geometry, the tuning c 1 = 0 and c 3 = 0 yields a new elliptic fibration (4.10) We can see that this elliptic fibration has, in addition to the zero-section at  indicating that the corresponding su(2) algebra is affected by the torsional section. Indeed, a quick glance at the codimension two loci reveals that the only enhancements along the su (2) divisor are at These geometric data hints towards an F-theory model with gauge symmetry SU (2) In the following, we will provide further evidence that F-theory compactified on J(Y b ) indeed gives rise to such a field theory. However, we will also see that the F-theory interpretation, in particular of the bisection geometry Y b , is much more obscured than in previously known examples.

F-theory on the Jacobian J (Y b )
We now wish to study the physics of the F-theory compactification on this Jacobian J(Y b ) in more detail. To this end, we first analyze the I 2 singularities above the codimension one locus In order to resolve the singularity we shall first perform a coordinate shift to locate the singularity in the fiber at the origin yielding the shifted Weierstrass model, which we also refer to as J(Y b ), The fibration again has two rational sections where the first is the zero-section and the latter the Z 2 torsional section.
We can resolve the singularity (4.17) by a blow-up (x, y, c 2 2 − 4 c 0 c 4 ; ζ) , (4.21) in the notation of [54]. Such a blow-up involves introduction of a new coordinate ζ and replacing x → x ζ , y → y ζ , c 2 2 − 4 c 0 c 4 → A ζ , (4.22) where [x : y : A] are now projective coordinates. Performing such a transformation in the hypersurface (4.19), followed by the proper transform, yields the resolved threefold,Ĵ(Y b ), described by the complete intersection  Table 4: Singular fibers and associated matter states of the blown-up Jacobian geometrŷ J(Y b ). The singularities over {c 0 } ∩ {c 4 } are terminal and thus not resolved.
The two exceptional divisors associated to the I 2 fiber are Thus, we conclude that F-theory on the partially resolved Jacobian geometryĴ(Y b ) has an SU (2)/Z 2 gauge symmetry without any localized matter. The gauge anomalies are straightforwardly checked to be canceled. To verify the gravitational anomaly cancellation, we can compute the Euler characteristic ofĴ(Y b ) with the procedure laid out in section 3.4. As explained in [63], the contribution of the terminal singularities are accounted for correctly if we treat the fibers as I 1 curves with χ top = 1. Thus we obtain the following contributions: (4.26) In the presence of terminal singularities, the Euler characteristic satisfies [63] where the second Betti number b 2 = 1 + h 1,1 (B) + rk(g) = 11 − K 2 B + rk(g) still satisfies the Shioda-Tate-Wazir formula. As spelled out in [63], the non-localized uncharged hypermultiplets are counted by whereas the localized uncharged hypermultiplets are counted by the points P ∈ B with ter- The only charged hypermultiplets come from the (charged) states of the su(2) adjoint representation, giving n c H = 2 + 12K 2 B (see table 4). With three vector multiplets from the su(2) gauge fields, the gravitational anomaly, In the Jacobian description, we can only see the massless su(2) gauge symmetry at the level of divisors. The presence of terminal singularities however signals some broken gauge symmetry under which the localized matter are charged. Since the Jacobian arises as the Jacobian of a geometry with a bisection, a natural proposal is that there is a U (1) broken to a Z 2 . Such a discrete remnant would manifest itself as a non-trivial torsion subgroup [73,74], which, however, is notoriously difficult to determine explicitly. We will refrain ourselves from attempting the necessary computation, and instead, in the next subsection, give evidence for the presence of the additional Z 2 discrete symmetry based on the consistency of Higgsing chains.

Matching the Spectrum via Higgsing
Any complex structure deformation of the geometry that modifies the gauge algebra and the multiplicities of the matter hypermultiplets corresponds to a field theoretic Higgsing that modifies the algebra and fields in the same way. Concretely, in our case, we have in mind a sequence of complex structure deformations with corresponding 6D field theory Higgsings: Field theory: The geometry J(Ŷ ) giving rise to an SU ( To match this Higgsed spectrum with that of F-theory on J(Y b ), we note that on J(Y b ), the geometric counting in the previous section does not distinguish between states of different 13 In this counting, the adjoint 3 of su(2) contains three hypermultiplets. Z 2 charge, e.g., su(2) singlets are all counted as uncharged hypermultiplets. In that case, the matching of the charged spectrum is straightforward, as it becomes just counting the number of su(2) adjoints after the Higgsing. From (4.33), we easily spot the 1 + 6K 2 B adjoint representations needed to match the geometric counting in table 4. Furthermore, we can also match the uncharged spectrum. Explicitly, we obtain additional uncharged hypermultiplets from the 1 1 and, importantly, also from the Z 2 charged adjoints 3 1 , where the state without Cartan charge is also an uncharged hypermultiplet in the geometric counting (4.30). Note that even though the 3 0 also contains a uncharged hypermultiplet, we do not have to include them in the counting of additional uncharged hypermultiplets, because they were already accounted for in the SU (2)/Z 2 × SU (2) model (3.31). In total, we then have additional uncharged hypermultiplets arising in the Higgsing (a), which together with the already present uncharged hypermultiplets (3.31) in the SU (2)/Z 2 × SU (2) phase precisely matches the number (4.30) computed geometrically forĴ(Y b ).

Higgsing step (b)
The above Higgsing leads to an SU (2)/Z 2 × Z 2 theory with the following charged spectrum su(2) ⊕ Z 2 Rep Multiplicity of hypermultiplets where the subscript denotes Z 2 charge. In order to higgs this to a theory with just a Z 2 discrete symmetry, corresponding to F-theory on J(Y Z 2 ), we again need a two-step Higgsing process.
First we give a vacuum expectation value to a hypermultiplet in the 3 0 representation, breaking Under this breaking, the Z 2 charged adjoints 3 1 decompose into singlets which are charged under both U (1) and Z 2 . Higgsing such a singlet then further breaks the gauge symmetry to a diagonal Z 2 . Explicitly, we obtain the following decomposition: It is straightforward to sum up the contributions to the singlets with Z 2 charge 1, which, with the multiplicities in (4.35), yields which is the number (4.5) of Z 2 charged singlets in the F-theory compactification on J(Y Z 2 ).

F-theory on the Bisection Model Y b
Thus far, we have analyzed the 6D F-theory compactification on the Jacobian fibration J(Y b ).
Requiring the two complex structure deformations (4.32), that connect J(Y b ) with the Z 2 torsion-enhanced Morrison-Park model J(Ŷ ) and with the standard Z 2 model J(Y Z 2 ), to be consistent with a field theoretic Higgsing process constrains the 6D theory to be an SU (2)/Z 2 × Z 2 gauge theory. This conclusion is further supported by the fibration structure of J(Y b ), which has a codimension one locus of I 2 fibers, a Z 2 torsional section, and terminal singularities in codimension two.
Based on observations made throughout the literature, one expects that an n-section geometry Y should give rise to the same F-theory compactification as its Jacobian J(Y ). Specifically, we know [63,73,74]  This situation persists in models where the 6D theory includes non-abelian gauge algebras g [33,73,[75][76][77][78]82]. In this setup, on the 5D Coulomb branch of either the Jacobian or the n-section geometry Y b , one must have rk(g) + #(indept. n-sections) independent U (1)s.
This however is not true for the case at hand! The deformation from Y Z 2 to Y b is a smooth deformation, and thus one has the same number, h 1,1 (Y Z 2 ), of divisors where not all deformation modes of the same non-abelian gauge divisor carry the same charges.
This conclusion is based on field theory arguments, which we believe should have a counterpart in geometry. In the Jacobian geometry J(Y b ), the discrete symmetry is encoded in torsional three-cycles [74] and is difficult to study directly. A better understanding of the bisection geometry may allow one to read off the discrete charges, including of non-localized adjoints, more directly.

Conclusions and Future Directions
In this paper, we have put forth a procedure to construct Weierstrass models of elliptic fibrations with a torsional Mordell-Weil generator that can be deformed to a free rational section. This procedure is exemplified by determining the Weierstrass model (2.15), which is birationally equivalent to any such elliptic fibration with a Z 2 torsional section. When this Weierstrass model is a Calabi-Yau threefold, the F-theory compactification to 6D yields a field theory with gauge group where one of the notable features is that the quotient does not act on every non-abelian factor of the gauge algebra. this complication may arise if we want to study analogous gauge enhancements of F-theory models with higher rank Mordell-Weil group or higher charge singlets. With the plethora of explicit F-theory models with rank ≥ 2 Mordell-Weil group [13,18,20,30,[83][84][85], an obvious extension would therefore be to solve the torsion condition directly in these constructions. For example, in models with multiple U (1)s, tuning one or more rational sections to be torsional might produce non-abelian gauge algebras with more intricate global structures. Furthermore, finding generic solutions for Mordell-Weil torsion Γ = Z 2 in general may be more than just an exercise in commutative algebra, and may give insights into new F-theory physics.
One may have hoped that by tuning the section to be torsional, the resulting gauge enhancement could also have produced higher dimensional representations, similar to constructions where one collides two or more sections [13][14][15]21]. There, the higher U (1) charges become the Cartan charges of higher dimensional non-abelian representations after the enhancement. On the other hand, when one tunes the section of the so-called U (1)-restricted Tate model [86] to be Z 2 torsional, one finds [10] that the charge 1 singlets (which are the only charged singlets of the restricted Tate model) enhances to su(2) adjoints with (highest) Cartan charge 2. Naively, one might have expected that the analogous tuning of a U (1) model with charge 2 singlets would result in an su(2) model with a 5 representation that has highest Cartan charge 4. However, our generic solution does not exhibit such higher dimensional representations, but instead a higher rank gauge group, whose breaking then yields the higher charged singlets. If we seek to restrict the generic solution so that the enhancement is rank preserving, or U (1) → SU (2)/Z 2 , then one finds that one must turn off the charge 2 locus in the U (1) model. This finding is consistent with the recent "swampland conjecture" [87], which forbids higher dimensional representations such as the 5 of su(2) in F-theory compactifications. It would be interesting to analyze if in models with U (1) charge > 2 [27,28,31], whose Morrison-Park model has "tall" sections and thus are non-Calabi-Yau [31], a torsional enhancement would generate novel higher dimensional representations.
Finally, we have studied a related deformation process of a Weierstrass model that arises as the Jacobian J(Y Z 2 ) of the bisection geometry Y Z 2 , whose F-theory compactification gives rise to a Z 2 symmetry [21,[73][74][75]77,78]. In section 4, we have seen that the deformed Jacobian J(Y b ) exhibits a Z 2 torsional section that comes with the massless SU (2)/Z 2 gauge symmetry.
At the same time, this Jacobian also has terminal singularities, which we argued field theoretically corresponds to localized singlets charged under a discrete Z 2 gauge symmetry. However, the field theory is rather peculiar, e.g., it contains su(2) adjoints with different Z 2 charges, which appear nevertheless not to be localized in geometry. From previous examples with discrete symmetries in the literature, one might have hoped to understand Z 2 charged matter better in the corresponding deformed bisection geometry Y b . However, the interpretation of the bisection geometry within the F-theory context only raises more questions. In particular, the genus-one fibration Y b lacks independent divisors giving rise to the Cartan of su(2) and the Kaluza-Klein U (1) that would be necessary to straightforwardly uplift M-theory on Y b to the F-theory defined on J(Y b ). Since there are by now a multitude of multi-section models in the literature that can consistently incorporate non-abelian gauge symmetries, it might possible, through a refined definition of F-theory on multi-section geometries such as Y b , to resolve some of these puzzles. We hope to shed some light on this issue in the future [88].
This geometric construction of the group law is depicted in figure 4. Adding a point A to itself can be seen as the limit of sending the point B to A, in which case the line through A and B becomes the tangent at A. For n = 2, 3, this relations can be visualized fairly easily, as depicted in figure 5. In particular, we see that a rational point with a vertical tangent is a Z 2 torsional point. Likewise, a rational inflection point is a Z 3 torsional point.

A.1 Z 2 Torsion
In the following, we would like to argue that on a smooth elliptic curve in the inhomogeneous , y(t)) = (t, ± t 3 + f t + g), where the sign depends on the branch. In that parametrization, it is also easy to compute the slope of the curve: Because the curve is smooth by assumption, we know that P W and its derivative dP inh W = −2 y dy + (3x 2 + f ) dx (A.4) cannot vanish simultaneously. In particular, this means that at a point with y = 0, the numerator 3x 2 + f in (A.3) cannot be zero. In turn, this means that the slope of elliptic curve (A.1) must be infinite at a point Q with y = 0. Thus, a tangent line at such a point is vertical and intersects the elliptic curve again only at infinity. From the above discussion (see also  curve is 3. Alternatively, one can also view the inflection point Q 3 as the limit of approaching two points Q and R that satisfy 2Q = R, see figure 5.

A.2 Z 3 Torsion
From the expression for the slope (A.3), we can easily determine its second derivative: Then, Q is an inflection point if it satisfies the condition The sign ambiguity simply reflects the fact that a smooth elliptic curve has two inflection points, or equivalently, the Weierstrass equation is symmetric under y ↔ −y; we will stick with + for definiteness. Furthermore, note that this relationship is derived from the inhomogeneous form (A.1) of the Weierstrass equation. To obtain an expression that is valid for an elliptic fibration, one needs to projectivize the (x, y)-plane to P 231 , i.e., including the appropriate factors of z. 14 Thus, the condition for a rational point Q in an elliptic fibration to be Z 3 torsional is

B Gauge Enhancement via Z 3 Torsion
In this appendix we explore the geometry where the section of the elliptic fibration is located a point of Z 3 torsion. After finding a simplified solution to the tuning condition and the associated F-theory spectrum, we study possible Higgsings back to a U (1) model and match the multiplicities. with (q 1 , q 4 ) and (q 2 , q 3 ) being coprime. But because A = c 2 1 is a complete square, q 1 and q 2 must combine into a square. As shown for instance in appendix A of [40], this has for solution q i = rη 2 i , i = 1, 2, where r is the factor common to q 1 and q 2 and (η 1 , η 2 ) are coprime. Similarly, (c 0 − t 2 ) 2 = q 2 q 4 = rη 2 2 q 4 implies that rq 4 must also be a square. However, as we have chosen a factorization such that (q 1 , q 4 ) are coprime, so must be (r, q 4 ), as r is a factor of q 1 . Therefore, both r and q 4 must be squares on their own, which in turn means that so must q 1 and q 2 , and we deduce that the generic solution to (B.4) is To determine the multiplicity of the matter, we employ the cancellation conditions of nonabelian anomalies. The multiplicities of adjoint matter is again found using formula (2.19): states, yields the following multiplicities of charged singlets: